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On Li-Yorke Measurable Sensitivity PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 143, Number 6, June 2015, Pages 2411–2426 S 0002-9939(2015)12430-6 Article electronically published on February 3, 2015 ON LI-YORKE MEASURABLE SENSITIVITY JARED HALLETT, LUCAS MANUELLI, AND CESAR E. SILVA (Communicated by Nimish Shah) Abstract. The notion of Li-Yorke sensitivity has been studied extensively in the case of topological dynamical systems. We introduce a measurable version of Li-Yorke sensitivity, for nonsingular (and measure-preserving) dynamical systems, and compare it with various mixing notions. It is known that in the case of nonsingular dynamical systems, a conservative ergodic Cartesian square implies double ergodicity, which in turn implies weak mixing, but the converses do not hold in general, though they are all equivalent in the finite measure- preserving case. We show that for nonsingular systems, an ergodic Cartesian square implies Li-Yorke measurable sensitivity, which in turn implies weak mixing. As a consequence we obtain that, in the finite measure-preserving case, Li-Yorke measurable sensitivity is equivalent to weak mixing. We also show that with respect to totally bounded metrics, double ergodicity implies Li-Yorke measurable sensitivity. 1. Introduction The notion of sensitive dependence for topological dynamical systems has been studied by many authors; see, for example, the works [3, 7, 10] and the references therein. Recently, various notions of measurable sensitivity have been explored in ergodic theory; see for example [2, 9, 11–14]. In this paper we are interested in formulating a measurable version of the topolog- ical notion of Li-Yorke sensitivity for the case of nonsingular and measure-preserving dynamical systems. In Section 2, we review some preliminary definitions and introduce the notion of Li-Yorke measurable sensitivity (also called Li-Yorke M-sensitivity); which is based on the topological notion of Li-Yorke sensitivity in [5]; and prove that in the conservative ergodic case it implies W-sensitivity introduced in [11]. In Section 3 we prove that if the Cartesian square is conservative ergodic (a condition stronger than weak mixing in the nonsingular case [1]), then it is Li-Yorke M-sensitive. Sec- tion 4 shows that for conservative ergodic nonsingular systems, an ergodic Cartesian square implies Li-Yorke M-sensitivity, which in turn implies weak mixing; a con- sequence of this is that in the finite measure-preserving case Li-Yorke sensitivity is equivalent to weak mixing. Section 5 studies scrambled sets. In Section 6 we consider conservative ergodic infinite measure-preserving transformations which are Received by the editors February 2, 2013 and, in revised form, August 8, 2013 and October 9, 2013. 2010 Mathematics Subject Classification. Primary 37A40; Secondary 37A05. Key words and phrases. Nonsingular transformation, measure-preserving, ergodic, Li-Yorke. This paper is based on research by the Ergodic Theory group of the 2011 SMALL summer research project at Williams College. Support for the project was provided by National Science Foundation REU Grant DMS - 0353634 and the Bronfman Science Center of Williams College. c 2015 American Mathematical Society 2411 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2412 J. HALLETT, L. MANUELLI, AND C. E. SILVA not W-measurably sensitive. The final sections study entropy and the existence of a W-measurably sensitive μ-compatible metric for any conservative ergodic trans- formation. 2. Preliminary definitions and measurable sensitivity A nonsingular dynamical system (X, S,μ,T) is a standard Borel space (X, S) with a σ-finite, nonatomic measure μ and a nonsingular endomorphism T : X → X (i.e., for all A ∈S, T −1(A) ∈ S and μA = 0 if and only if μ(T −1A)=0).We sometimes take T to be measure-preserving or the measure space to be finite. We sometimes suppress S and write (X, μ, T). We say that T is conservative if for all A of positive measure there exists n>0 such that μ(T −nA ∩ A) > 0. A set A is positively invariant if A ⊂ T −1(A) and invariant if T −1(A)=A.IfT is conservative and A is positively invariant, then A is invariant mod μ (henceforth all equalities will be interpreted as mod μ). A transformation T is ergodic if whenever A is invariant, then μ(A)=0orμ(X \ A) = 0. A nonsingular transformation T is weakly mixing if whenever f is an L∞ function such that f ◦T = zf for z ∈ C,then f is constant a.e. If T × T is ergodic, then T is weakly mixing, but the converse, while true in the finite measure-preserving case, does not hold in general [1]. We consider metrics d on X. We assume throughout that these are Borel mea- surable on X × X and bounded by 1. We say a metric d on X is μ-compatible if μ assigns positive measure to nonempty, open d-balls [11, 14]. It follows from [14, Lemma 1.1] that the topology generated by d is separable. Thus open sets are measurable as they are countable unions of balls. The notion of measurable sensitivity was introduced in [14]. Definition 2.1. We say a nonsingular dynamical system (X, μ, T)ismeasurably sensitive if for every isomorphic mod 0 dynamical system (X1,μ1,T1)andμ1- compatible metric d on X1 there exists a δ>0 such that for all x ∈ X1 and ε>0thereisann ∈ N such that { ∈ n n } μ1 y Bε(x):d(T1 (x),T1 (y)) >δ > 0. This definition was refined in [11]. Definition 2.2. Let (X, μ, T) be a nonsingular dynamical system and d a μ- compatible metric on X. WesaythesystemisW-measurably sensitive with respect to d if there is a δ>0 such that for each x ∈ X lim sup d(T nx, T ny) >δ n→∞ for a.e. y ∈ X. The system is W-measurably sensitive if it is W-measurably sensitive with respect to each μ-compatible metric d. Proposition 7.2 in [11] shows that for conservative ergodic transformations these two notions are equivalent. Requiring that the condition hold for each x ∈ X and a.e. y ∈ X is equivalent to requiring that it hold for a.e. pair (x, y) ∈ X2 (Proposition A.1 in [11]), a notion called pairwise sensitivity introduced in [9]. The following classification result is proved in [11, Theorem 7.1]. Theorem 1. Let (X, μ, T) be a conservative ergodic nonsingular dynamical sys- tem. Then T is W-measurably sensitive or T is isomorphic mod 0 to an invertible minimal uniformly rigid isometry on a Polish space. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON LI-YORKE MEASURABLE SENSITIVITY 2413 The proof of this theorem shows that the isometric metric is μ-compatible. The following are two technical propositions from [11] that are needed for our work. The proofs follow those in the aforementioned paper and are included for completeness. Proposition 2.1. Suppose T is a nonsingular transformation. If for almost every pair (x, y) ∈ X × X there exists n ≥ 0 such that d(T nx, T ny) ≥ δ, then for almost ∈ × n n ≥ every pair (x, y) X X we have lim supn→∞ d(T x, T y) δ. Proof. Let Z(N,x)= y ∈ X : ∃n, d(T n(T N x),Tny) ≥ δ . Then by hypothesis there exists a full measure set A such that μ(Z(N,x)c)=0for each x ∈ A and for all N ∈ N.Let Y (N,x)={y ∈ X : ∃n>N,d(T nx, T ny) ≥ δ} . Note that Y (N,x)=T −N (Z(N,x)). Since Z(N,x) has full measure and T is nonsingular, we see that Y (N,x) also has full measure. Hence Y (N,x) N>0 also has full measure. But this says for each x ∈ A and almost every y ∈ X that n n ≥ lim supn→∞ d(T x, T y) δ. The proof of the next proposition follows from the same arguments as above. Proposition 2.2. Suppose T is a nonsingular transformation. If for almost every pair (x, y) ∈ X × X there exists n ≥ 0 such that d(T nx, T ny) ≤ δ, then for almost n n every pair (x, y) ∈ X × X we have lim infn→∞ d(T x, T y) ≤ δ. The notion of W-measurable sensitivity adapts the notion of sensitivity to ini- tial conditions from topological dynamics to the measurable case. In topological dynamics there is also the notion of Li-Yorke sensitivity, in which points are not only required to separate but also to come back together. We give the definitions of topological Li-Yorke sensitivity as in [5]. We recall the following definition from topological dynamics. Definition 2.3. Let (X, d, T) be a topological dynamical system. A pair (x, y)is said to be proximal if lim inf d(T nx, T ny)=0. n→∞ Akin and Kolyada introduced the following in [5]. Definition 2.4. Let (X, d, T) be a topological dynamical system. We call the system Li-Yorke sensitive if there exists an ε>0 such that every x ∈ X is a limit of points y ∈ X such that the pair (x, y) is proximal but whose orbits are at least ε apart at arbitrarily large times. In this paper we consider the measure-theoretic analogue of Li-Yorke sensitivity. Definition 2.5. Let (X, μ, T) be a nonsingular dynamical system and d a μ- compatible metric on X. We say that a pair (x, y)isaLi-Yorke pair if n n n n lim infn→∞ d(T x, T y) = 0 and lim supn→∞ d(T x, T y) > 0.
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